Z-Score Calculator

Standardize a value as a signed number of standard deviations from its mean.

Drag & drop or upload an image or PDF

Enter a raw score, mean, and standard deviation, such as x = 85, mean = 70, SD = 10

Provide the raw value x, the matching mean, and a positive standard deviation. Say whether these are population or sample summaries.

Try an example

A z-score is a standardized location

A z-score tells how many standard deviations a value sits above or below its mean. Positive scores are above the mean, negative scores are below it, and zero is exactly at the mean.

Standardizing puts measurements from different scales into comparable units. It does not make the original data normal, and a percentile interpretation needs a distribution model rather than the z-score formula alone.

How to use the z-score calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

Raw score to standard score

Subtract the mean to get the signed distance, then divide by the standard deviation to express that distance in standardized units.

z = \frac{x - μ}{σ}

A reliable way to work through it

Match the value to its summary

Use a mean and standard deviation computed for the same variable, group, and units as the raw score.

Keep the sign of the difference

The sign shows which side of the mean contains the value and must remain visible through the division.

Interpret without overreaching

Describe distance from the mean directly. Add percentiles or normal tail areas only when a normal model is stated.

Worked example

Find the z-score for x = 85 when the mean is 70 and the standard deviation is 10

x - μ = 85 - 70 = 15

The raw score is 15 units above the mean.

z = \frac{85 - 70}{10} = \frac{15}{10} = 1.5

Divide the signed difference by the positive standard deviation.

x = μ + z σ = 70 + 1.5 (10) = 85

Reverse the standardization to check the result.

The z-score is 1.5, so the raw value is 1.5 standard deviations above the mean.

Common mistakes to check

Reversing the subtraction

Use raw value minus mean. Reversing the order changes the sign and the interpretation.

Using standard error by accident

A descriptive z-score for one observation divides by the distribution's standard deviation, not the standard error of a sample mean.

Treating z as a percentile

A z-score is a standardized distance. Converting it to a percentile requires a stated distribution, commonly a normal model.

Standard Deviation CalculatorMeasure how far values spread from the mean, with the sample or population denominator shown clearly.Mean Median Mode CalculatorCompare three measures of center and see how sorting, frequencies, and outliers affect them.Probability CalculatorTranslate an event question into the right rule without assuming independence or double-counting overlap.Permutation and Combination CalculatorCount ordered arrangements and unordered selections, with the order decision made before the factorials.

Questions students ask

What does a negative z-score mean?

It means the raw value is below the mean. The magnitude tells how many standard deviations separate them.

Can a z-score be greater than 3?

Yes. Z-scores are not restricted to a fixed interval, although large magnitudes may be unusual under some distribution models.

Does a z-score require normally distributed data?

No for standardizing a value. Normality is needed only when using the z-score to read probabilities or percentiles from the standard normal distribution.

How do I convert a z-score back to a raw value?

Use x = mean + z times standard deviation, keeping all three quantities from the same distribution.